Yes, £800 fun money is a lot for someone saving hard for a deposit. I would look to halve it or save a further £500 at least.
Within reason, it's good to be a bit flexible about area. You don't have to absolutley love it....that's a luxury most first time buyers can't really afford...and it's not worth going for HTB and its disadvantages if you can get somewhere without it, which still ticks most of your boxes.
Don't be shy with your parents. Sharing with them how much you've saved (they will be impressed, especially if you cut the fun money and show you're willing to really cut back for a while) and the different prices and deposits required for HTB compared to a conventional route, might show them that just a couple of extra grand from them could make the difference. You don't need to ask for money or expect it, but many parents who have some money would be more than glad to give a couple of grand to help their child. I know that not everyone can do this, but you don't sound in this position. I think if you'll have dsaved £20-25k, then you can still hold your head up and feel you've saved hard, even if given an extra couple of grand to get you over the threshold. If it makes the difference between HTB and traditional route, it really would be great.
Op, you sound in a good position. HTB is clearly a possibility in your mind, but you also have the option to avoid it....you've saved a good chunk already, plan to save more, plus have the capacity to save even more via your fun fund being cut down, which you can easily do. Given all these positives, you really don't sound like you NEED to go for HTB. For some people, it really is the only option as they have very limited deposit and little chance of saving much more in a reasonable timeframe...but you're not in this position. It's just a question of whether you're tempted by the low deposit you can afford now and the shiny new and possibly the location, which are short term benefits with longer term costs associated with them, vs the longer saving period and less shiny place and possibly location, but longer term benefits and cheaper costs in the end.